Hierarchical structure of the family of curves with maximal genus verifying flag conditions

Fix integers such that and , and let be the set of all integral, projective and nondegenerate curves of degree in the projective space , such that, for all , does not lie on any integral, projective and nondegenerate variety of dimension and degree . We say that a curve satisfies the flag condition if belongs to . Define where denotes the arithmetic genus of . In the present paper, under the hypothesis , we prove that a curve satisfying the flag condition and of maximal arithmetic genus must lie on a unique flag such as , where, for any , denotes an integral projective subvariety of of degree and dimension , such that its general linear curve section satisfies the flag condition and has maximal arithmetic genus . This proves the existence of a sort of a hierarchical structure of the family of curves with maximal genus verifying flag conditions.

     

A bound on the geometric genus of projective varieties verifying certain flag conditions

Fix integers and let be the set of all integral, projective and nondegenerate varieties of degree and dimension in the projective space , such that, for all , does not lie on any variety of dimension and degree . We say that a variety satisfies a flag condition of type if belongs to . In this paper, under the hypotheses , we determine an upper bound , depending only on , for the number , where denotes the geometric genus of . In case and , the study of an upper bound for the geometric genus has a quite long history and, for , and , it has been introduced by Harris. We exhibit sharp results for particular ranges of our numerical data . For instance, we extend Halphen's theorem for space curves to the case of codimension two and characterize the smooth complete intersections of dimension in as the smooth varieties of maximal geometric genus with respect to appropriate flag condition. This result applies to smooth surfaces in . Next we discuss how far is from and show a sort of lifting theorem which states that, at least in certain cases, the varieties of maximal geometric genus must in fact lie on a flag such as , where denotes a subvariety of of degree and dimension . We also discuss further generalizations of flag conditions, and finally we deduce some bounds for Castelnuovo's regularity of varieties verifying flag conditions.

     

On certain remarkable curves of genus five

The aim of this note is twofold. First to show the existence of genus five curves having exactly twenty four Weierstrass points, which constitute the set of fixed points of three distinct elliptic involutions on them. Second to characterize these curves, in fact we prove that all such curves are bielliptic double cover of Fermat's quartic.     

On certain curves of genus three in characteristic two

We study curves of genus 3 over algebraically closed fields of characteristic 2 with the canonical theta characteristic totally supported in one point. We compute the moduli dimension of such curves and focus on some of them which have two Weierstrass points with Weierstrass directions towards the support of the theta characteristic. We answer questions related to order sequence and Weierstrass weight of Weierstrass points and the existence of other Weierstrass points with similar properties. – Dedicated to the treasured memory of our coauthor, Paulo Henrique Viana     

The genus of space curves

LetC be a curve contained in ℙ _k ³ (k of any characteristic), which is locally Cohen-Macaulay, not contained in a plane and of degreed. We prove thatp _a (C)≤≤((d−2)(d−3))/2. Moreover we show existence of curves with anyd, p _a satisfying this inequality and we characterize those curves for which equality holds.

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Sunto Si dimostra che seC⊃ℙ _k ³ (k di caratteristica qualunque) é una curva, non piana, localmente Cohen-Macaulay, di gradod, allorap _a (C)≤((d−2)(d−3))/2. Si mostra che questa limitazione è ottimale, e si classificano le curve di genere aritmetico massimale.

     

Geometry of curves in generalized flag varieties

The current paper is devoted to the study of integral curves of constant type in generalized flag varieties. We construct a canonical moving frame bundle for such curves and give a criterion when it turns out to be a Cartan connection. Generalizations to parametrized curves, to higher-dimensional submanifolds and to parabolic geometries are discussed.     

Plane curves of genus p and degree 2p as projections of space curves of the same degree

We characterize plane curves Γ of genus p and degree 2p with respect to the possibility of obtaining them as projections of space curves C′ of the same degree. When Γ is hyperelliptic, we link this characterization with the configuration of the singularities of Γ and with the position of C′ on certain scrolls. Supported by the M.U.R.S.T. of the Italian Government     

Higher rank BN-theory for curves of genus 4

Higher rank Brill–Noether theory is completely known for curves of genus ≤ 3. In this paper, we investigate the theory for curves of genus 4. Some of our results apply to curves of arbitrary genus.     

The arithmetic of certain quartic curves

We give a sufficient condition using the Brauer?CManin obstruction under which certain quartic curves have no rational points. Using this sufficient condition, we construct two families of genus one quartic curves violating the Hasse principle explained by the Brauer?CManin obstruction.     

Arithmetically cohen-macaulay curves inP 4 of degree 4 and genus 0

We show that the arithmetically Cohen-Macaulay (ACM) curves of degree 4 and genus 0 in P⁴ form an irreducible subset of the Hilbert scheme. Using this, we show that the singular locus of the corresponding component of the Hilbert scheme has dimension greater than 6. Moreover, we describe the structures of all ACM curves ofHilb ^4m+1 (P⁴). Supported in part by the Norwegian Research Council (Matematisk Seminar). Supported in part by école Normale Supérieure and the Nansen Fund. This article was processed by the author using the Springer-Verlag T_EX PJourlg macro package 1991.     

Euler characteristic of certain affine flag varieties

The purpose of this note is to prove, as Lusztig stated, that the Euler characteristic of the variety of Iwahori subalgebras containing a certain nil-elliptic elementn _t istc^l wherel is the rank of the associated finite type Lie algebra. The author's research is supported in part by a National Science Foundation postdoctoral fellowship.     

The moduli space of curves of genus two covering elliptic curves

We consider the moduli spaceS _n of curvesC of genus 2 with the property:C has a “maximal” mapf of degreen to an elliptic curveE. Here, the term “maximal” means that the mapf∶C→E doesn't factor over an unramified cover ofE. By Torelli mapS _n is viewed as a subset of the moduli spaceA ₂ of principally polarized abelian surfaces. On the other hand the Humbert surfaceH _Δ of invariant Δ is defined as a subvariety ofA ₂(C), the set of C-valued points ofA ₂. The purpose of this paper is to releaseS _n withH _Δ.     

On certain domains in cycle spaces of flag manifolds

The affine homogeneous space associated to a real semi-simple Lie group G with maximal compact subgroup K contains a number of naturally defined{\it G}-invariant neighborhoods of its real points which are of interest from various points of view. Here the universal Iwasawa domain is introduced from the point of view of incidence geometry and certain of its properties are derived, e.g., it is Stein, Kobayashi hyperbolic and contains the domain introduced by Akhiezer and Gindikin which is now known to be equivalent to the maximal domain of definition of the adapted complex structure associated to the Killing metric in the tangent bundle . One of the main goals of the paper is to develop methods which lead to a better understanding of the Wolf domain of cycles in an open G-orbit D in a flag manifold . The key is the Schubert domain which is defined by Schubert cycles of complementary dimension to the cycles. These are defined by a Borel subgroup containing an Iwasawa factor AN and consequently and are closely related. Received: 28 May 2001 / Revised version: 19 November 2001 / Published online: 23 May 2002     

Modular curves of genus 2

We prove that there are exactly genus two curves defined over such that there exists a nonconstant morphism defined over and the jacobian of is -isogenous to the abelian variety attached by Shimura to a newform . We determine the corresponding newforms and present equations for all these curves.

     

The orbifold cohomology of moduli of genus 3 curves

In this work we study the additive orbifold cohomology of the moduli stack of smooth genus g curves. We show that this problem reduces to investigating the rational cohomology of moduli spaces of cyclic covers of curves where the genus of the covering curve is g. Then we work out the case of genus g =  3. Furthermore, we determine the part of the orbifold cohomology of the Deligne–Mumford compactification of the moduli space of genus 3 curves that comes from the Zariski closure of the inertia stack of ${\mathcal{M}3}$ .     

The generalized Verschiebung map for curves of genus 2

Let C be a smooth curve, and M _r (C) the coarse moduli space of vector bundles of rank r and trivial determinant on C. We examine the generalized Verschiebung map induced by pulling back under Frobenius. Our main result is a computation of the degree of V ₂ for a general C of genus 2, in characteristic p > 2. We also give several general background results on the Verschiebung in an appendix.This paper was partially supported by fellowships from the National Science Foundation and Japan Society for the Promotion of Sciences.